There could be common denominators not in , then how do you know which one to pick?

I think when you say "discrete" you mean "concrete".

Yes, I mean this specific examples using these specific numbers. As far as the GCD goes, the book did make the statement that "[the] GCD is already assumed as defined and there is no need to decompose 24 or 49 into primes in Q[Sqrt(3)]]".

So can someone show me how to find the GCD of those two numbers within the respective field mentioned?

Yes, I mean this specific examples using these specific numbers. As far as the GCD goes, the book did make the statement that "[the] GCD is already assumed as defined and there is no need to decompose 24 or 49 into primes in Q[Sqrt(3)]]".

So can someone show me how to find the GCD of those two numbers within the respective field mentioned?

I think what your book means by that is that we assume the ring of integers in question is a GCD domain and in particular a Euclidean domain (and even more in particular, a norm-Euclidean domain).

And I think we use the definition that an integer g is the gcd of two integers x,y if and only if for for all integers d: d divides x and d divides y implies that d divides g.

Using this definition I think the gcd will be unique up to multiplication by units.

I've been looking at the two subheadings "Euclidean domains" and "Unique factorization of quadratic integers" here at Wikipedia. (And also the subheadings before that.) And it seems in general that the Euclidean algorithm produces the desired result whether or not at each step we choose a remainder as small as possible (here, smallness given by the norm), as long the sequence of remainders is strictly decreasing.

At any rate, 49 = 24 * 2 + 1. And 2 = 1 * 2 + 0. So I think the gcd is just 1, same as in .

Hope you don't mind my constantly saying "I think" and "it seems". Truth is I have little experience in this, so while I believe I've come to correct conclusions, I'm not as sure as I would be with experience.